An Observer Standing On The Top Of A Lighthouse 150 M Above Sea Level Watches A Ship Sailing Away. As He Observes, The Angle Of Depression Of The Ship Changes From 50° To 30°.Determine The Distance Traveled By The Ship During The Period Of Observation.

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Introduction

The problem of determining the distance traveled by a ship as observed from a lighthouse is a classic application of trigonometry. In this article, we will use the concept of right-angled triangles and trigonometric ratios to calculate the distance traveled by the ship. We will also explore the relationship between the angle of depression and the distance traveled by the ship.

The Problem

An observer standing on the top of a lighthouse 150 m above sea level watches a ship sailing away. As he observes, the angle of depression of the ship changes from 50° to 30°. We need to determine the distance traveled by the ship during the period of observation.

Understanding the Concept of Angle of Depression

The angle of depression is the angle between the horizontal and the line of sight to the object. In this case, the object is the ship. As the ship sails away, the angle of depression changes from 50° to 30°. This means that the observer's line of sight to the ship is changing, and we need to take this into account when calculating the distance traveled by the ship.

Using Trigonometry to Calculate the Distance Traveled

To calculate the distance traveled by the ship, we can use the concept of right-angled triangles. We can draw a right-angled triangle with the lighthouse as the vertex, the ship as the opposite side, and the line of sight to the ship as the hypotenuse. We can then use the trigonometric ratios to calculate the distance traveled by the ship.

Calculating the Distance Traveled Using the Tangent Ratio

We can use the tangent ratio to calculate the distance traveled by the ship. The tangent ratio is defined as the ratio of the opposite side to the adjacent side. In this case, the opposite side is the distance traveled by the ship, and the adjacent side is the height of the lighthouse.

Let's denote the distance traveled by the ship as x. We can then write the equation:

tan(50°) = 150 / x

We can solve for x by multiplying both sides of the equation by x:

x = 150 / tan(50°)

x ≈ 102.64 m

This is the distance traveled by the ship when the angle of depression is 50°.

Calculating the Distance Traveled Using the Tangent Ratio (Again)

We can use the same method to calculate the distance traveled by the ship when the angle of depression is 30°. We can write the equation:

tan(30°) = 150 / x

We can solve for x by multiplying both sides of the equation by x:

x = 150 / tan(30°)

x ≈ 173.21 m

This is the distance traveled by the ship when the angle of depression is 30°.

Calculating the Distance Traveled by the Ship

Now that we have calculated the distance traveled by the ship when the angle of depression is 50° and 30°, we can calculate the distance traveled by the ship during the period of observation. We can use the formula:

distance traveled = (distance traveled at 50° + distance traveled at 30°) / 2

distance traveled ≈ (102.64 + 173.21) / 2

distance traveled ≈ 137.93 m

This is the distance traveled by the ship during the period of observation.

Conclusion

In this article, we used the concept of right-angled triangles and trigonometric ratios to calculate the distance traveled by a ship as observed from a lighthouse. We calculated the distance traveled by the ship when the angle of depression is 50° and 30°, and then used these values to calculate the distance traveled by the ship during the period of observation. The result shows that the distance traveled by the ship is approximately 137.93 m.

References

  • [1] "Trigonometry" by Michael Corral, 2019.
  • [2] "Geometry" by I.M. Gelfand, 1962.
  • [3] "Mathematics for the Nonmathematician" by Morris Kline, 1967.

Further Reading

  • [1] "Trigonometry: A First Course" by Harry L. Alder, 1969.
  • [2] "Geometry: A Comprehensive Introduction" by Dan Pedoe, 1970.
  • [3] "Mathematics: A Human Approach" by Harold R. Jacobs, 1970.

Glossary

  • Angle of depression: The angle between the horizontal and the line of sight to the object.
  • Tangent ratio: The ratio of the opposite side to the adjacent side in a right-angled triangle.
  • Right-angled triangle: A triangle with one angle equal to 90°.
  • Hypotenuse: The side opposite the right angle in a right-angled triangle.

Introduction

In our previous article, we explored the problem of determining the distance traveled by a ship as observed from a lighthouse. We used the concept of right-angled triangles and trigonometric ratios to calculate the distance traveled by the ship. In this article, we will answer some of the most frequently asked questions related to this problem.

Q: What is the angle of depression, and how does it affect the distance traveled by the ship?

A: The angle of depression is the angle between the horizontal and the line of sight to the object. In this case, the object is the ship. As the ship sails away, the angle of depression changes, and this affects the distance traveled by the ship. The angle of depression is an important factor in calculating the distance traveled by the ship.

Q: How do I calculate the distance traveled by the ship using the tangent ratio?

A: To calculate the distance traveled by the ship using the tangent ratio, you need to use the following formula:

tan(θ) = opposite side / adjacent side

where θ is the angle of depression, and the opposite side is the distance traveled by the ship.

Q: What is the formula for calculating the distance traveled by the ship when the angle of depression is 50°?

A: The formula for calculating the distance traveled by the ship when the angle of depression is 50° is:

x = 150 / tan(50°)

where x is the distance traveled by the ship.

Q: What is the formula for calculating the distance traveled by the ship when the angle of depression is 30°?

A: The formula for calculating the distance traveled by the ship when the angle of depression is 30° is:

x = 150 / tan(30°)

where x is the distance traveled by the ship.

Q: How do I calculate the distance traveled by the ship during the period of observation?

A: To calculate the distance traveled by the ship during the period of observation, you need to use the following formula:

distance traveled = (distance traveled at 50° + distance traveled at 30°) / 2

Q: What is the distance traveled by the ship during the period of observation?

A: The distance traveled by the ship during the period of observation is approximately 137.93 m.

Q: What are some real-world applications of this problem?

A: This problem has many real-world applications, such as:

  • Calculating the distance traveled by a ship or a plane
  • Determining the height of a building or a mountain
  • Calculating the distance between two points on a map

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

  • Not using the correct formula for calculating the distance traveled by the ship
  • Not taking into account the angle of depression
  • Not using the correct values for the tangent ratio

Q: How can I practice solving this problem?

A: You can practice solving this problem by:

  • Using online calculators or software to calculate the distance traveled by the ship
  • Creating your own problems and solving them
  • Working with a partner or a group to solve the problem

Conclusion

In this article, we answered some of the most frequently asked questions related to the problem of determining the distance traveled by a ship as observed from a lighthouse. We hope that this article has been helpful in clarifying any doubts you may have had about this problem.

References

  • [1] "Trigonometry" by Michael Corral, 2019.
  • [2] "Geometry" by I.M. Gelfand, 1962.
  • [3] "Mathematics for the Nonmathematician" by Morris Kline, 1967.

Further Reading

  • [1] "Trigonometry: A First Course" by Harry L. Alder, 1969.
  • [2] "Geometry: A Comprehensive Introduction" by Dan Pedoe, 1970.
  • [3] "Mathematics: A Human Approach" by Harold R. Jacobs, 1970.

Glossary

  • Angle of depression: The angle between the horizontal and the line of sight to the object.
  • Tangent ratio: The ratio of the opposite side to the adjacent side in a right-angled triangle.
  • Right-angled triangle: A triangle with one angle equal to 90°.
  • Hypotenuse: The side opposite the right angle in a right-angled triangle.